In this report, we build monthly Black-Litterman portfolios for six Fama-Fench portfolios (FF6).  These portfolios are named after the research methodologies of two University of Chicago professors, Eugene Fama and Kenneth French. They extended the single-factor Capital Asset Pricing Model (CAPM) to include two additional factors: the difference in returns between small \& large stocks, and the difference between returns of high book-to-market stocks \& low book-to-market stocks.These factors are thought to account for the tendenancy for small cap stocks to outperform large cap stocks and for value stocks to outperform growth stocks.

Our portfolio\footnote{The vwret specified in the project data files is not log return. Hence, we convert all the values to log returns at the beginning.} of stocks traded on NYSE, AMEX, and NASDAQ from January 1992 to Jan 2012. Each of our equity portfolios are characterized by the size (market equity) and book-to-market ratio:

\begin{description}
\itemsep0em
\item[SMLO]: small size and low book-to-market ratio
\item[SMME]: small size and medium book-to-market ratio
\item[SMHI]: small size and high book-to-market ratio
\item[BILO]: big size and low book-to-market ratio
\item[BIME]: big size and medium book-to-market ratio
\item[BIHI]: big size and high book-to-market ratio
\end{description}

We will use the short names to refer to the portfolio throughout the rest of the report. As we can see in Figure. \ref{fig:markowitz_enigma}, Markowitz's mean-variance optimization, using plugin estimates $\hat{\mu}$ and $\hat{\Sigma}$, leads to unbalanced portfolios. One special concern is that Markowitz's M-V optimization does not consider the asset market capitalizations weights, shown in Figure. \ref{fig:cap_weights}. For example, the allocation gives really high weights to the SMME portfolio, which has very small market capitalization. Moreover, Markowitz M-V optimization is very sensitive to the input, as shown in Figure. \ref{fig:mv_transition}. The weights (we use the absolute values for clear plots here) allocated for each portfolio, are highly dynamic across the time, yet this is caused by small updates to the estimated returns.  This is undesirable for an optimal portfolio allocation strategy. Hence, we decide to use the Black-Litterman model to optimize our portfolio allocations\cite{intuition2002}. 

\begin{figure}[htb]
\centering
\includegraphics[width=7.5cm]{../results/markowitz_enigma.png}
\caption{Potfolio weights by Markowitz M-V Optimization using plug-in estimates $\mu$ and $\sigma$}
\label{fig:markowitz_enigma}
\end{figure}

\begin{figure}[htb]
\centering
\includegraphics[width=7.5cm]{../results/cap_weights.png}
\caption{Market capital weights of the 6 portfolios}
\label{fig:cap_weights}
\end{figure}

\begin{figure}[htb]
\centering
\includegraphics[width=7.5cm]{../results/mv_abs_transition.png}
\caption{Transition Map of Allocated Weights (absolute) of Each Portfolio Across time}
\label{fig:mv_transition}
\end{figure}

To build the B-L portfolio, we first estimate $\Pi$ (equilibrium risk premium) using reverse optimization. This serves as a neutral starting point for the optimization process. Then, we construct absolute views for the (next period) returns for all 6 portfolios using various time series models (ARMA, AR-GARCH, and ARMA-VIX). The predicted view vectors are then taken by the Black-Litterman model to calculate the posterior $\mu^{BL}$ and $\Sigma^{BL}$.

Using the B-L estimation of $\mu^{BL}$ and $\Sigma^{BL}$, we compute the optimal allocation weights using mean-variance optimization (with short selling allowed). To evaluate our models, we track the cumulative returns and Sharpe ratios over time, and compare them with the traditional M-V portfolio using plug-in estimates. We also compare the performance of different view vectors, estimated using different models. In the experiment, we consider two different way to select training data: expanding window and rolling window. The expanding window uses all the available data in the past, after initializing our models with 5 years of data. But, since stocks are affected by more recent events, we also consider a rolling window of the most recent 5 years of data.  In the end, we also propose an approach to also consider the portfolio turnover rate, which may incur high monthly transaction costs in the allocation strategy.

We will now outline a summary of the rest of our report. In Section \ref{sec:prima}, we will describe our calculation of equilibrium risk prima. In Section \ref{sec:view}, we will explain the details of the forecast models used to construct the view vector for the B-L model. The process of B-L optimization and its results are discussed in Section \ref{sec:bl}. The experimental results for all different approaches are discussed in Section \ref{sec:eval}. Last, we discuss extending our strategy to consider the effects of turn over rate in Section \ref{sec:turnover}.